*A geometric sequence with a common ratio may be defined by a first order difference equation.*

## The Equation

A geometric sequence where there is a common ratio of can be defined by a first order difference equation of the form:

Where is the common ratio

is an increasing sequence

is a decreasing sequence

is a sequence alternating between positives and negatives

## First Order Difference Equations Defining Geometric Sequences

For a first order difference equation to define a geometric sequence, there must be a common ratio and no addition or subtraction of terms. For first order difference equations, the geometric common ratio is the pro numeral .

### Example 1

**Which of the following first order difference equations defines a geometric sequence?**

**a) **

This first order difference equation does define an (increasing) geometric sequence as it has a common ratio of 4.

**b) **

This first order difference equation does define a geometric sequence (that alternates between positive and negative numbers) as it has a common ratio of -2.

**c) **

This first order difference equation does *not* define a geometric sequence as there is an addition of 2.

## Expressing Geometric Sequences as First Order Difference Equations

If we have the general form of a geometric sequence, we can use to find and to find .

Be careful! in the form of refers to the first term and in refers to the common ratio.

### Example 2

**Express the geometric sequence where as a first order difference equation.**

In , the common ratio () is 3 and the first term () is 4.

Therefore, for the first order difference equation , and .

Now we just have to substitute into the equation.

We can also express a geometric sequence as a first order difference equation if we only have terms. We just have to calculate the common ratio and find the first term.

### Example 3

**Express the following geometric sequence as a first order difference equation.**

We are told that it is a geometric sequence so we don’t need to test for a common ratio.

So our common ratio is 3.

We can see that the first term is 8.

Therefore, the first order difference equation is given by

#### See also

Recognition of Geometric Sequences

Finding the Terms of a Geometric Sequence

First Order Difference Equations

Generating the Terms of a Sequence Defined by a First Order Difference Equation

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